Can infinite joins/meets be viewed as special cases of a limit with respect to some topology?

Question

Let $L$ denote a complete lattice. Is there a topology on $L$ such that infinite joins and meets can be viewed as a special kind of "limit"?

Answer 1

Any partially ordered set has a topology generated by the open intervals $(a, b) = \{ c : a < b < c \}$ and half-open intervals $(a, \infty) = \{ c : a < c \}, (-\infty, a) = \{ c : c < a \}$. Infinite suprema and infima are limits with respect to this topology. (I would call this the order topology except that this term appears to be reserved for the special case of totally ordered sets for reasons I don't understand.)

But there is an even better way to understand infinite joins and meets that also uses the word "limit"! Namely, they are categorical (co)limits, thinking of a partially ordered set as a category where $a \le b$ means that there is a unique arrow from $a$ to $b$.